\setlength\subproofhorizspace{1.6em}
\begin{logicproof}{1}
  \exists x \; \lnot P(x)                       & prem.\\
  \forall x \; \lnot Q(x)                       & prem.\\
  \begin{subproof}
	  \llap{$x_0\enspace \;$} \lnot P(x_0)        & ass.\\
    \lnot Q(x_0)                                & $\forall \mathrm{e}$ 2\\
    \lnot P(x_0) \land \lnot Q(x_0)             & $\land \mathrm{i}$ 3,4\\
    \exists x \; (\lnot P(x) \land \lnot Q(x))  & $\exists \mathrm{i}$ 5
  \end{subproof}
  \exists x \; (\lnot P(x) \land \lnot Q(x))    & $\exists \mathrm{e}$ 3-6
\end{logicproof}